Abstract Measurements of anisotropic flow coefficients with two- and multi-particle cumulants for inclusive charged particles in Pb-Pb collisions at $$ \sqrt{s_{\mathrm{NN}}}=5.02 $$ s N N = 5.02 and 2.76 TeV are reported in the pseudorapidity range |η| < 0.8 and transverse momentum 0.2 < p T < 50 GeV/c. The full data sample collected by the ALICE detector in 2015 (2010), corresponding to an integrated luminosity of 12.7 (2.0) μb−1 in the centrality range 0-80%, is analysed. Flow coefficients up to the sixth flow harmonic (v 6) are reported and a detailed comparison among results at the two energies is carried out. The p T dependence of anisotropic flow coefficients and its evolution with respect to centrality and harmonic number n are investigated. An approximate power-law scaling of the form v n (p T) ∼ p T n/3 is observed for all flow harmonics at low p T (0.2 < p T < 3 GeV/c). At the same time, the ratios v n /v m n/ m are observed to be essentially independent of p T for most centralities up to about p T = 10 GeV/c. Analysing the differences among higher-order cumulants of elliptic flow (v 2), which have different sensitivities to flow fluctuations, a measurement of the standardised skewness of the event-by-event v 2 distribution P(v 2) is reported and constraints on its higher moments are provided. The Elliptic Power distribution is used to parametrise P(v 2), extracting its parameters from fits to cumulants. The measurements are compared to different model predictions in order to discriminate among initial-state models and to constrain the temperature dependence of the shear viscosity to entropy-density ratio.